Properties

Label 68.1.f.a
Level 68
Weight 1
Character orbit 68.f
Analytic conductor 0.034
Analytic rank 0
Dimension 2
Projective image \(D_{4}\)
CM disc. -4
Inner twists 4

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 68 = 2^{2} \cdot 17 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 68.f (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.033936420859\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{4}\)
Projective field Galois closure of 4.2.19652.1
Artin image size \(32\)
Artin image $C_4\wr C_2$
Artin field Galois closure of 8.0.1257728.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + i q^{2} - q^{4} + ( -1 - i ) q^{5} -i q^{8} + i q^{9} +O(q^{10})\) \( q + i q^{2} - q^{4} + ( -1 - i ) q^{5} -i q^{8} + i q^{9} + ( 1 - i ) q^{10} + q^{16} - q^{17} - q^{18} + ( 1 + i ) q^{20} + i q^{25} + ( 1 + i ) q^{29} + i q^{32} -i q^{34} -i q^{36} + ( -1 - i ) q^{37} + ( -1 + i ) q^{40} + ( 1 - i ) q^{41} + ( 1 - i ) q^{45} -i q^{49} - q^{50} + ( -1 + i ) q^{58} + ( -1 + i ) q^{61} - q^{64} + q^{68} + q^{72} + ( 1 + i ) q^{73} + ( 1 - i ) q^{74} + ( -1 - i ) q^{80} - q^{81} + ( 1 + i ) q^{82} + ( 1 + i ) q^{85} + ( 1 + i ) q^{90} + ( -1 - i ) q^{97} + q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 2q^{5} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{5} + 2q^{10} + 2q^{16} - 2q^{17} - 2q^{18} + 2q^{20} + 2q^{29} - 2q^{37} - 2q^{40} + 2q^{41} + 2q^{45} - 2q^{50} - 2q^{58} - 2q^{61} - 2q^{64} + 2q^{68} + 2q^{72} + 2q^{73} + 2q^{74} - 2q^{80} - 2q^{81} + 2q^{82} + 2q^{85} + 2q^{90} - 2q^{97} + 2q^{98} + O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/68\mathbb{Z}\right)^\times\).

\(n\) \(35\) \(37\)
\(\chi(n)\) \(-1\) \(i\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
1.00000i
1.00000i
1.00000i 0 −1.00000 −1.00000 1.00000i 0 0 1.00000i 1.00000i 1.00000 1.00000i
55.1 1.00000i 0 −1.00000 −1.00000 + 1.00000i 0 0 1.00000i 1.00000i 1.00000 + 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
4.b Odd 1 CM by \(\Q(\sqrt{-1}) \) yes
17.c Even 1 yes
68.f Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(68, [\chi])\).